Definition – Sum – Math-reference.net

Contents

1. Introduction
2. Definitions

Introduction

Here we introduce many things which come under sum. For example, when adding up multiple elements of a set, several types of sums such as infinite sums, partial sums etc, as well as some definitions which may not themselves be sums, but include the word sum as part of a well accepted definition.

Definition – Upper sum of a subset of $\mathbb{R}$

Let $f$ be a bounded function defined on $[a, b]$ , and let $X = \{x_0, ..., x_n\}$ be a partition of $[a, b]$ . Finally, let

$\begin{array}{ll} C_i &= sup\{f(x)~|~x\in[x_i, x_{i-1}]\} \\\\ c_i &= inf\{f(x)~|~x\in[x_i, x_{i-1}]\} \end{array}$

We define the upper and lower sums, $U(f, X)$ and $\L(f,X)$ , as

$\begin{array}{ll} U(f, P) = \sum\limits_{i=0}^n C_i(x_i-x_{i-1}) \\\\ L(f, P) = \sum\limits_{i=0}^n c_i(x_i-x_{i-1}) \end{array}$

Where $U(f, P)$ is the upper sum, and $L(f, P)$ is the lower sum.